Maths.CLIL analisi matematica, le derivate

Derivatives

Maths.CLIL

• The difference quotient
• Notation
• Right derivative

Derivatives

Maths.CLIL

• The difference quotient
• Notation
• Right derivative

Learn the basics

Let us consider a continuous function y = f(x) and two of its points, P(x₀; f(x₀)) and Q(x₀ + h; f(x₀ + h)). The difference quotient is defined as

Δy/Δx = (f(x₀ + h) - f(x₀))/h

It represents the slope of the secant line connecting P and Q.

As we take Q closer and closer to P, the slope of the secant line connecting Q and P is getting closer and closer to the slope of the tangent line to the function at P. The derivative of a function at that point represents the slope of the tangent line. Its definition is

f'(x₀) = lim_Δx→0 Δy/Δx = lim_h→0 (f(x₀ + h) - f(x₀))/h.

The derivative exists if the limit exists and has a finite value.

Notation

We can use different notations to denote the derivative of a function f with respect to x.

The first notation is to write f^′(x) for the derivative of the function f(x). This functional notation was introduced by Lagrange, based on Isaac Newton’s ideas. The prime mark in f^′(x) denotes that f^′(x) is derived from f(x). We can read it as “first derivative of f(x)” or “f prime of x”.

The second is ^df/_dx. This is called Leibniz’s notation, and it refers to the instantaneous rate of change of y with respect to x. It was introduced by Gottfried Wilhelm Leibniz, one of the discoverers of calculus.

We can also use the symbol D to indicate the operation of differentiation.